 reserve x for Real,
    p,k,l,m,n,s,h,i,j,k1,t,t1 for Nat,
    X for Subset of REAL;

theorem Th10:
  for i,j being natural Number holds
  i <= j implies ex k st j = i + k
proof
  let i,j be natural Number;
A0: j is Nat by TARSKI:1;
  defpred P[Nat] means i <= $1 implies ex k st $1 = i + k;
A1: for j st P[j] holds P[j+1]
  proof
    let j such that
A2: i <= j implies ex k st j = i + k;
A3: now
      assume i <= j;
      then consider k such that
A4:   j = i + k by A2;
      i + k + 1 = i + (k + 1);
      hence thesis by A4;
    end;
A5: now
      assume i = j + 1;
      then j + 1 = i + 0;
      hence thesis;
    end;
    assume i <= j + 1;
    hence thesis by A3,A5,Th8;
  end;
A6: P[0]
  proof
    assume
A7: i <= 0;
    take 0;
    thus thesis by A7,Th2;
  end;
  for j holds P[j] from NatInd(A6,A1);
  hence thesis by A0;
end;
